The Queue-Number of Posets of Bounded Width or Height

Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width $2$ has queue-number at most $2$, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width $w$ have queue-number at most $3w-2$ while any planar poset with $0$ and $1$ has queue-number at most its width.Comment: 14 pages, 10 figures, Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Mixed Linear Layouts of Planar Graphs

A $k$-stack (respectively, $k$-queue) layout of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing (non-nested) edges with respect to the vertex ordering. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed $1$-stack $1$-queue layout in which every edge is assigned to a stack or to a queue that use a common vertex ordering. We disprove this conjecture by providing a planar graph that does not have such a mixed layout. In addition, we study mixed layouts of graph subdivisions, and show that every planar graph has a mixed subdivision with one division vertex per edge.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Functional properties of edible insects: a systematic review

: Consumption of edible insects has been widely suggested as an environmentally sustainable substitute for meat to reduce GHG emissions. However, the novel research field for edible insects rely on the content of bioactive ingredients and on the ability to induce a functional effect in humans. The goal of this manuscript was to review the available body of evidence on the properties of edible insects in modulating oxidative and inflammatory stress, platelet aggregation, lipid and glucose metabolism and weight control. A search for literature investigating the functional role of edible insects was carried out in the PUBMED database using specific keywords. A total of 55 studies, meeting inclusion criteria after screening, were divided on the basis of the experimental approach: in vitro studies, cellular models/ex vivo studies or in vivo studies. In the majority of the studies, insects demonstrated the ability to reduce oxidative stress, modulate antioxidant status, restore the impaired activity of antioxidant enzymes and reduce markers of oxidative damage. Edible insects displayed anti-inflammatory activity reducing cytokines and modulating specific transcription factors. Results from animal studies suggest that edible insects can modulate lipid and glucose metabolism. The limited number of studies focused on the assessment of anticoagulation activity of edible insects make it difficult to draw conclusions. More evidence from dietary intervention studies in humans is needed to support the promising evidence from in vitro and animal models about the functional role of edible insects consumption

Stack and Queue Layouts via Layered Separators

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Hierarchical Partial Planarity

In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to construct layouts of these graphs in which the readability of an edge is proportional to its importance, that is, more important edges have fewer crossings. We formalize this problem and study the case in which there exist three different degrees of importance. We give a polynomial-time testing algorithm when the graph induced by the two most important sets of edges is biconnected. We also discuss interesting relationships with other constrained-planarity problems.Comment: Conference version appeared in WG201

Splitting Clusters To Get C-Planarity

In this paper we introduce a generalization of the c-planarity testing problem for clustered graphs. Namely, given a clustered graph, the goal of the S PLIT-C-P LANARITY problem is to split as few clusters as possible in order to make the graph c-planar. Determining whether zero splits are enough coincides with testing c-planarity. We show that S PLIT-C-P LANARITY is NP-complete for c-connected clustered triangulations and for non-c-connected clustered paths and cycles. On the other hand, we present a polynomial-time algorithm for flat c-connected clustered graphs whose underlying graph is a biconnected seriesparallel graph, both in the fixed and in the variable embedding setting, when the splits are assumed to maintain the c-connectivity of the clusters

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure

Prostaglandin PGE2 at very low concentrations suppresses collagen cleavage in cultured human osteoarthritic articular cartilage: this involves a decrease in expression of proinflammatory genes, collagenases and COL10A1, a gene linked to chondrocyte hypertrophy

Suppression of type II collagen (COL2A1) cleavage by transforming growth factor (TGF)-β2 in cultured human osteoarthritic cartilage has been shown to be associated with decreased expression of collagenases, cytokines, genes associated with chondrocyte hypertrophy, and upregulation of prostaglandin (PG)E2 production. This results in a normalization of chondrocyte phenotypic expression. Here we tested the hypothesis that PGE2 is associated with the suppressive effects of TGF-β2 in osteoarthritic (OA) cartilage and is itself capable of downregulating collagen cleavage and hypertrophy in human OA articular cartilage. Full-depth explants of human OA knee articular cartilage from arthroplasty were cultured with a wide range of concentrations of exogenous PGE2 (1 pg/ml to 10 ng/ml). COL2A1 cleavage was measured by ELISA. Proteoglycan content was determined by a colorimetric assay. Gene expression studies were performed with real-time PCR. In explants from patients with OA, collagenase-mediated COL2A1 cleavage was frequently downregulated at 10 pg/ml (in the range 1 pg/ml to 10 ng/ml) by PGE2 as well as by 5 ng/ml TGF-β2. In control OA cultures (no additions) there was an inverse relationship between PGE2 concentration (range 0 to 70 pg/ml) and collagen cleavage. None of these concentrations of added PGE2 inhibited the degradation of proteoglycan (aggrecan). Real-time PCR analysis of articular cartilage from five patients with OA revealed that PGE2 at 10 pg/ml suppressed the expression of matrix metalloproteinase (MMP)-13 and to a smaller extent MMP-1, as well as the proinflammatory cytokines IL-1β and TNF-α and type X collagen (COL10A1), the last of these being a marker of chondrocyte hypertrophy. These studies show that PGE2 at concentrations much lower than those generated in inflammation is often chondroprotective in that it is frequently capable of selectively suppressing the excessive collagenase-mediated COL2A1 cleavage found in OA cartilage. The results also show that chondrocyte hypertrophy in OA articular cartilage is functionally linked to this increased cleavage and is often suppressed by these low concentrations of added PGE2. Together these initial observations reveal the importance of very low concentrations of PGE2 in maintaining a more normal chondrocyte phenotype

Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

Upward Three-Dimensional Grid Drawings of Graphs

A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar